Mach number: RPN vs. AOS comparison


I evaluated the Mach-number equation given by r.d. baertschiger below:

M= sqrt[ 5 [({ [( 1 + 0.2 [ 350/661.5 ]^2 )^3.5 -1 ] [ 1- ( 6.875^1E-6 ) 25500 ]^-5.2656 } +1 )^0.286 -1 ] ]

using a number of diiferent machines.


-- RPN (12C, 15C, 17Bii, 28S, 48G, 49G):

I got the correct answer (0.835724536) every time within one minute of work, once I saw how the expressions nested (match the braces "{}" by multiplying two long results.)

-- Non-symbolic AOS with precedence (20S, Casio fx-3600p):

I got the correct answer on the first try with somewhat more time, when entering the expression exactly as written, which utilizes precedence. Doing the ()'s slows one down a little.

-- Symbolic AOS with precedence (28S, 48G, 49G):

I got the correct answer on the first try, but required more time, due to all the "()". This method gives no intermediate results, however, so the final answer must almost be accepted on faith, unless one scrutinizes the symbolic expressions after-the-fact.

-- Symbolic AOS without precedence (17Bii):

Serious difficulties getting the correct answer, with repeated failures. Extra ()'s must be inserted at the right time to handle lack of precedence, and if it wasn't done, it's difficult to fix.

-- Non-symbolic AOS without precedence (10B):

Repeated failures required completely starting over, and ()'s were shifted. Horrible!

-- Equation writers/editors (48G, 49G, 17Bii):

Very cumbersome; expressions were hard to read. Many failures. 48G equation writer could not keep up with user input.


-- RPN:

Probably best for correctly solving this type of problem, because you see all intermediate results, and you know what they represent. Error recovery is easy. Of course, a strategy for evaluation must be devised beforehand, but RPN users are good at it.

-- Non-symbolic AOS with precedence:

Also useable, but the intermediate results produced by ")" are less obvious. (How many operations were performed, and what were they?) The user can enter the problem exactly as written, but it must be written according to the rules of precedence. "Fixing" the calculation midstream is nearly impossible.

Instead of entering the experssion left-to-right, a quasi-RPN approach can be emulated with liberal use of "=" and storage registers.

-- Symbolic AOS and equation writers/editors:

Allows the user to see the complete expression, but not the intermediate results upon evaluation. I wouldn't trust 'em.

- AOS without precedence:

Leave 'em for the business people!


I couldn't resist and had a go using a few calculators.

I found RPN on my 11C and 42S easy and quick once I had written the equation by hand with vertical structure and superscripts.

I then turned to my 71B and typed it in very quickly, but got an error message because it gives ^ precidence over unary- This was fixed with brackets around the (-5.2656)


This gives M=0.835724535179

Other results were:

17BII 0.835724535179

42S 0.83572453518

11C 0.835724536

Overall I found each method equally slow to set up and debug and about the same speed after debug. I like both RPN and good algebraic calcs!


is that a bug? something like 2^-2, do you have to write 2^(-2)?


No, it is not a bug, but the 71B implements operator precedence with ^ first and then unary- next, this means that it tries to evaluate 2^- before pulling in the number and gets an undefined result, so brackets are essential! Note thay 2^+5 also gives an error, but 2^5 is fine.

This is a strange order and I would like to hear from anyone who has an idea as to why the 71B operator precedence is like this.

Precedence of Operators (from 71B Quick Ref Guide).

The table below lists HP-71 operators in their order of

precedence. Where an expression contains two or more

operators having the same level of precedence, those oper-

ators will be evaluated in the left-to-right order in
which they occur within the expression.

Performed first:

(...) (Nested parentheses are evaluated from the
inside out.)
Functions (such as SIN, RND, etc.)


unary +, unary -, NOT

*, /, \, DIV, %

+, -, &

<, =, >, #, ?, <=, >=, <>



Performed last.


Hi Karl,

I got pretty similar results, except that I was not as fast!. However, I found the 20s to be just as efficient and easy as RPN--actually I got the right answer on the 20S first (And I am a died in the wool RPN guy!).

But on the symbolic type systems, namely 48GX equation writer: that beast is as useless as teets on a bore-hog! SLOOOOOW and cumbersome and a paint to edit in--it was actually much easier to create an algebraic object on the stack. I do not see any good in that equation writer thing for an equation like this! I wonder if that-there ALG48 or ERABLE or METAkernel whathaveyou is worth it.

I also tried writing the equation into my 32sii equation space--boy did that use a lot of memory! And I still got the wrong answer---don't know why!




Quick (VERY quick) editors -numeric & symbolic matrices, equations, text, graphics, etc- with cut, copy and paste..
I actually think that the 48 is MUCH better with MK. I wouldn't use it without MK today...


Forgive me, what is the MetaKernel? An emulator of another calculator for a calculator?


MetaKernel is some sort of cool alternative graphic environment.

But I do have an emulator for another calculator running on a caclulator--it emulates the 41c series and runs on the 48G series---its very cool!

Regarding M-K, I downloaded it, but cannot figire out how to turn it on. I also cannot figire out how to get a lot of other stuff "on" such as XCELL (a spreadsheet) ERABLE (a computer algebra thing) and ERABLE (another computer algebra thing.).




MetaKernel: port1
Erable: part in port0 and other parts in port2
Alg48: port2




Noooooo: it's (in some way) a better OS for the 48GX



If it's not too much trouble, can you briefly summarize how so?

I will admit that using a 48G reminds me a bit of the old days of VMS on mainframes and their clunky little text editors... almost. Nothing really is as bad as that (VMS and all that)!


Here you have a review and condensed version of the documentation but the best is to install and test...

Edited: 7 Oct 2003, 2:24 a.m.


I just wanted to point out that the review whose link Raul provided speaks of Meta Kernel being a 128K ROM card and says that it costs about US$90. That was true at the time it was written, but the authors have since released it as a free download. It's available at the same place as the review,, at this link: You need a 128K RAM card to use it, which restricts its use to a 48GX. (This is an excellent use for one of the inexpensive Klotz 128K cards.) It fills the entire card, so port 1 has to be dedicated to Meta Kernel, but it's well worth it. With MK in slot 1 and another RAM card (in my case a 2MB card) in slot 2 the 48GX is a whole new machine.


Ah, razzberrys!

I only have up to a G+, 128K native. But I'll review the files and see for myself what you guys were talking about anyway.

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